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Some Useful Cases in Constraint Relation

In cases where distance between two points is always fixed, we can say the relative velocity of one point of an object with respect to any other point of the same object in the direction of the line joining them will always remain zero, as their separation always remains constant.

Application of constraint relation

Case I
Consider a rod of length l resting on a wall and the floor (Fig. 10). Its lower end A is pulled towards left with a constant velocity u. Result of this end B starts moving down along the wall. Let us find the velocity of the other end B downward when the rod makes an angle θ with the horizontal.
Fig. 10
Here the distance between the points A and B of the rod always remains constant, thus the two points must have same velocity components in the direction of their line joining i.e., along the length of the rod.
If point B is moving down with velocity vB, its component along the length of the rod is vB sin θ. Similarly, the velocity component of point A along the length of rod is v cos θ. Thus, we have
vB sinθ = u cosθ
or vB = u cotθ
Case II
Consider a ball of mass m1 and a block of mass m2 are joined together with an inextensible string (Fig. 11). The ball can slide on a smooth horizontal surface. If v1 and v2 are the respective speeds of the ball and the block, let us find the constraint relation between the two.
Fig. 11
As length of the string is constant, hence the velocity of end points along the string is same. Obviously, from Fig. 12, 35529.png.
Fig. 12

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